364 research outputs found
On deciding stability of multiclass queueing networks under buffer priority scheduling policies
One of the basic properties of a queueing network is stability. Roughly
speaking, it is the property that the total number of jobs in the network
remains bounded as a function of time. One of the key questions related to the
stability issue is how to determine the exact conditions under which a given
queueing network operating under a given scheduling policy remains stable.
While there was much initial progress in addressing this question, most of the
results obtained were partial at best and so the complete characterization of
stable queueing networks is still lacking. In this paper, we resolve this open
problem, albeit in a somewhat unexpected way. We show that characterizing
stable queueing networks is an algorithmically undecidable problem for the case
of nonpreemptive static buffer priority scheduling policies and deterministic
interarrival and service times. Thus, no constructive characterization of
stable queueing networks operating under this class of policies is possible.
The result is established for queueing networks with finite and infinite buffer
sizes and possibly zero service times, although we conjecture that it also
holds in the case of models with only infinite buffers and nonzero service
times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002)
272--293] and uses the so-called counter machine device as a reduction tool.Comment: Published in at http://dx.doi.org/10.1214/09-AAP597 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results
Our model is a constrained homogeneous random walk in a nonnegative orthant
Z_+^d. The convergence to stationarity for such a random walk can often be
checked by constructing a Lyapunov function. The same Lyapunov function can
also be used for computing approximately the stationary distribution of this
random walk, using methods developed by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly is an undecidable problem: no algorithm can exist to achieve this task.
We then prove that computing large deviation rates for this model is also an
undecidable problem. We extend these results to a certain type of queueing
systems. The implication of these results is that no useful formulas for
computing stationary probabilities and large deviations rates can exist in
these systems
The achievable region method in the optimal control of queueing systems : formulations, bounds and policies
Cover title.Includes bibliographical references (p. 44-48).Supported in part by a Presidential Young Investigator Award, with matching funds from Draper Laboratory. DDM-9158118Dimitris Bertsimas
The achievable region method in the optimal control of queueing systems : formulations, bounds and policies
Cover title.Includes bibliographical references (p. 44-48).Supported in part by a Presidential Young Investigator Award, with matching funds from Draper Laboratory. DDM-9158118Dimitris Bertsimas
Fluid and Diffusion Limits for Bike Sharing Systems
Bike sharing systems have rapidly developed around the world, and they are
served as a promising strategy to improve urban traffic congestion and to
decrease polluting gas emissions. So far performance analysis of bike sharing
systems always exists many difficulties and challenges under some more general
factors. In this paper, a more general large-scale bike sharing system is
discussed by means of heavy traffic approximation of multiclass closed queueing
networks with non-exponential factors. Based on this, the fluid scaled
equations and the diffusion scaled equations are established by means of the
numbers of bikes both at the stations and on the roads, respectively.
Furthermore, the scaling processes for the numbers of bikes both at the
stations and on the roads are proved to converge in distribution to a
semimartingale reflecting Brownian motion (SRBM) in a -dimensional box,
and also the fluid and diffusion limit theorems are obtained. Furthermore,
performance analysis of the bike sharing system is provided. Thus the results
and methodology of this paper provide new highlight in the study of more
general large-scale bike sharing systems.Comment: 34 pages, 1 figure
A mathematical programming approach to stochastic and dynamic optimization problems
Includes bibliographical references (p. 46-50).Supported by a Presidential Young Investigator Award. DDM-9158118 Supported by matching funds from Draper Laboratory.Dimitris Bertsimas
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